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  1. Home > Articles & Issues >
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  3. Bigraphical arrangem ...
Conference paper

Bigraphical arrangements

Sam Hopkins ORCID (1), David Perkinson (2)
(1) Massachusetts Institute of Technology
(2) Reed College
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Imported on
November 21, 2016
Published on
December 31, 2013
Last modified on
March 31, 2025
Volume
DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)
Proceedings
DOI
10.46298/dmtcs.2398
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Bigraphical arrangements

Sam Hopkins ORCID (1), David Perkinson (2)
(1) Massachusetts Institute of Technology
(2) Reed College
Abstract
We define the bigraphical arrangement of a graph and show that the Pak-Stanley labels of its regions are the parking functions of a closely related graph, thus proving conjectures of Duval, Klivans, and Martin and of Hopkins and Perkinson. A consequence is a new proof of a bijection between labeled graphs and regions of the Shi arrangement first given by Stanley. We also give bounds on the number of regions of a bigraphical arrangement. The full version of this paper is forthcoming in the $\textit{Transactions of the American Mathematical Society}$
Keywords
  • [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]
  • [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]
  • hyperplane arrangements
  • parking functions
  • abelian sandpile model
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